Arriving in Time: Estimation of English Auctions with a Stochastic
Number of Bidders∗†
José J. Canals-Cerdá‡
Jason Pearcy§
January 2, 2015
Abstract
We develop a new econometric approach for estimation of second-price ascending-bid auctions with a stochastic number of bidders. Our empirical framework considers the arrival process
of new bidders as well as the distribution of bidders’ valuations of objects being auctioned. By
observing the timing of bidder arrival the model is identified even when the number of potential
bidders is stochastic and unknown. The relevance of our approach is illustrated with an empirical application using a unique dataset of art auctions on eBay. Our results suggest a higher
impact of sellers’ reputation on bidders’ valuations than previously reported in cross-sectional
studies but the impact of reputation on bidder arrival is largely insignificant. Interestingly, a
seller’s reputation impacts not only the actions of the bidders but the actions of the seller as
well. In particular, experience and a good reputation increase the probability of a seller posting
items for sale on longer-lasting auctions which we find increases the expected revenue for the
seller.
JEL Classification: C51, C72, D44, L11, L14.
Keywords: English Auctions, Internet markets, Structural estimation.
∗
For useful comments we thank Robert McNown, the editors Arthur Lewbel, Keisuke Hirano, and Jonathan
Wright, an Associate editor, two referees, seminar participants at the University of Colorado, and participants at
the following conferences: Conference in Tribute to Jean-Jacques Laffont, Toulouse, May 2005; The 75th Southern
Economic Association Meetings; The XXX Simposio de Analisis Economico; The Midwest Economic Association
Meetings, April 2006; The Summer Meetings of the Econometric Society, June 2006; and The Midwest Econometrics
Group Meetings, October 2006. Excellent editorial assistance was provided by Katrina Beck. Excellent research
assistance was provided by David Donofrio, Tyson Gatto, and especially Woong Tae Chung and Kelvin Tang. We are
grateful to Kristen Stein, artist and eBay power seller, for answering many questions on the functioning of eBay and
on bidders’ and artists’ behavior. Canals-Cerdá gratefully acknowledges funding from a CARTSS small grant from
the University of Colorado. These are the views of the authors, and should not be attributed to any other person or
organization, including the Federal Reserve Bank of Philadelphia.
†
c
Author Posting. José
J. Canals-Cerdá and Jason Pearcy 2015. This is the authors’ version of the work. It is
posted here for personal use, not for redistribution. The definitive version was published in the Journal of Business
and Economic Statistics, Vol. 31, No. 2, 2013, pp. 125-135. http://dx.doi.org/10.1080/07350015.2012.747825
‡
Federal Reserve Bank of Philadelphia, 10 Independence Mall, Philadelphia, PA 19106 ([email protected])
§
Department of Agricultural Economics and Economics, Montana State University, Bozeman, MT 59717-2920
([email protected])
1
Introduction
We develop a new econometric approach for estimation of second-price ascending-bid auctions with
a stochastic number of bidders. This approach is relevant for eBay auctions and is also of value for
other auction settings. Our empirical framework considers the arrival process of new bidders as well
as the distribution of bidders’ valuations for objects being auctioned. A fully parametric partial
likelihood model is developed allowing for unobserved heterogeneity and flexible bidder behavior.
The partial likelihood is based on observing the arrival of new bidders and the transaction price.
We assume that the timing of a bidder’s first bid represents her time of arrival to the auction. Our
model of bidder behavior is based on the assumptions made by Haile and Tamer (2003). These
assumptions, along with eBay’s proxy bidding system with low bid increments, imply that the
highest bid from the second highest bidder (the transaction price) equals that bidder’s valuation.
By observing the timing of bidder arrival in addition to the submitted bids, the model is identified
even when the number of potential bidders is stochastic and unknown. The relevance of our
approach is illustrated with an empirical application using a unique dataset of art auctions on
eBay.
This research contributes to the growing literature on structural estimation of auctions while
paying special attention to the specific features of the eBay marketplace. We view our approach as
complementary to the work of others as our approach is based on a non-overlapping set of assumptions required for identification and a novel estimation approach. Our research also contributes to
the growing literature on online markets (Lucking-Reiley, 2000).
A significant body of work has emerged which uses a structural approach, especially for firstprice auctions, but also for second-price ascending auctions and other types of auction formats.
Surveys include Hendricks and Paarsch (1995), Perrigne and Vuong (1999), and Hendricks and
Porter (2000). Researchers have used the structural approach for many purposes: to provide estimates of optimal reserve prices (Paarsch 1997), to analyze the effects of bidder collusion (Baldwin,
Marshall and Richard, 1997), to measure the extent of the winner’s curse and simulate seller revenue under different reserve prices (Bajari and Hortaçsu 2003), to analyze the effects of congestion
(Canals-Cerdá, 2005) or advertising in an online auctions market (Canals-Cerdá, 2006).
Most of the existing structural econometric models of second-price ascending auctions have
1
been inspired by the button auction model of Milgrom and Weber (1982). This model has a unique
dominant-strategy equilibrium where each bidder submits a bid equal to their valuation of the
object being auctioned. A prevalent approach, proposed by Donald and Paarsch (1996), assumes
that the highest bid of each one of the auction participants, except the winner of the auction,
equals their true valuation (see also Paarsch, 1997; Hong and Shum, 2003; Bajari and Hortaçsu,
2003). Alternatively, some papers require only that the winning bid equals the valuation of the
second-highest bidder and do not use the information from losing bids (Paarsch, 1992; Baldwin
et al., 1997; Haile, 2001). In a remarkable paper, Haile and Tamer (2003) use information on the
highest bid from each bidder, but they do not require any bid to equal the bidder’s true valuation
of the object being auctioned. We use the same assumptions regarding bidder behavior as Haile
and Tamer (2003) and show that these assumptions along with eBay’s proxy bidding result in the
transaction price being equal to the valuation of the second highest bidder when there are two or
more bidders. Information regarding other bids is used to the extent to which these bids change
the standing price which in turn influences observed bidder arrival.
Two main features distinguish eBay/Internet auctions from the button auction model. First,
not all bidders necessarily bid their true valuation and secondly, the actual number of bidders is
observed but the potential number of bidders is stochastic and unknown. (Athey and Haile (2007)
distinguish potential bidders from actual bidders and discuss the two deviations from the button
auction model mentioned here. See Section 4 and Section 6.3 of their paper.) Applying button
auction methods to estimate Internet auction models is problematic because identification results
for almost all ascending auction environments require the observation of the potential number of
bidders.
Numerous studies make use of eBay data. Song (2004), Bajari and Hortaçsu (2003; 2004),
Lucking-Reiley (2000) and others provide descriptions of the eBay auction format. Similar to us,
Song (2004) uses a structural framework to estimate an independent private value model with
eBay data. Both this paper and Song’s paper are primarily interested in the estimation of the
distribution of private values, F (v), when the number of potential bidders is unknown. Song
estimates her model using semi-parametric techniques, while the model presented in this paper is
fully parametric. While it’s possible to develop a semi-parametric version of our model, we leave
this for future work. The primary difference between our approach and Song’s is a non-overlapping
2
set of assumptions for identification.
Song identifies the distribution of private values from the observation of two order statistics
independent of the potential number of bidders. Because on eBay the highest bid is not observed,
Song relies on the highest bid from the second and third highest bidders. In addition, Song places
restrictions on the data in order to increase the chances of satisfying the identification assumptions
in her paper. These restrictions require the exclusion of a significant proportion of auctions. In
fact, if we had to restrict our data to auctions with three or more bidders in which a bid greater
than the third-highest bid was submitted not earlier than two days before the end of the auction,
as is done in Song (2004), we would need to disregard more than 85% of auctions.
Table 1 presents three stylized examples of eBay bidding behavior and illustrates the drawbacks
to the traditional button auction model and Song’s (2004) approach. Each auction contains four
potential bidders with increasing valuations. Each bidder arrives at a different time and submits a
proxy bid which influences the current price. In each auction, the number of actual bidders is less
than the potential number of bidders. This happens because potential bidders with lower valuations
arrive to the auction at a time when the current minimum acceptable bid is above their valuation.
The first auction is consistent with the button auction for actual bidders, but inconsistent if one
considers the set of potential bidders.
Another study similar to ours is by Ackerberg, Hirano and Shahriar (2006). Their study evaluates eBay’s buy-it-now feature and uses a structural framework to estimate a model using eBay
data. The process of bidder arrival as well as the distribution of bidders’ valuations is modeled
using similar behavioral assumptions. Their estimation technique employs a partial likelihood approach combined with moment simulation while our estimation technique is based only on a partial
likelihood approach.
The rest of the paper proceeds as follows. Section 2 presents a structural model of eBay auctions
and describes the econometric methodology. In Section 3 we consider an empirical application of our
methodology utilizing a unique dataset of art auctions on eBay. A model is estimated to evaluate
the impact of sellers’ reputation on bidder arrival and on bidders’ valuations. Our results suggest a
higher impact of sellers’ reputation on bidders’ valuations than previously reported in cross-sectional
studies but the impact of reputation on bidder arrival is largely insignificant. We also analyze which
factors influence the sellers choice of auction length. Interestingly, a seller’s reputation impacts not
3
only the actions of the bidders but the actions of the seller as well. In particular, experience
and a good reputation increase the probability of a seller posting items for sale on longer-lasting
auctions. Finally, we conduct a counter-factual analysis simulating what would happen if seven-day
auctions were instead listed as ten-day auctions. This exercise highlights the value of our structural
approach. Our analysis indicates that if seven-day auctions were instead listed as ten-day auctions
sellers’ overall revenue would increase by 9%. Section 4 concludes. More descriptive information
about the dataset and eBay auctions are included in a supplementary appendix.
2
A Structural Model of eBay Auctions
A detailed description of eBay auctions is found in Song (2004), Bajari and Hortaçsu (2003; 2004),
Lucking-Reiley (2000) and others, and is not repeated here. We model eBay auctions as independent
private value (IPV), ascending-bid, second-price auctions subject to some specific rules. The seller
of the object being auctioned sets the duration of the auction, T , and the starting value, s0 .
Each potential bidder, n, assigns a value, vn , to the object being auctioned. Each value, vn , is a
independent realization from a distribution F (v) with density f (v). Bidders know and care only
about their own valuation.
A proxy bidding system is used where bids, b, are submitted electronically at any time t within
the [0, T ] time interval. The bid history at time t, denoted B(t), includes previous bid amounts with
the corresponding time of bid submission. At each time t, the price of the auction, denoted s(t), is
set at the current second highest bid plus a minimum increment. New bids arrive sequentially at
any time during the [0, T ] interval. The auction begins with s(0) = s0 and ends with a transaction
price of s(T ).
Any new bid has to surpass s(t) by a minimum increment in order to be recorded. The value
of the minimum increment in eBay auctions varies with the standing bid, s(t). The minimum
increment is $0.05 for standing bids under $1.00 and increases up to $100.00 for standing bids
above $5000.00. The average increment in our empirical application is $1 which is slightly more
than 1% of the transaction price. Throughout the paper we assume that the minimum increment
is negligible and do not mention it to avoid additional notation. Bid increments are discussed in
the Supplementary Appendix.
4
For a particular auction ij (artist i auctioning object j), Nij is the number of potential bidders
which is the number of bidders who would have chosen to bid in an otherwise identical sealed
bid auction for which any bid amount is acceptable. Mij represents the total number of bids
from observed bidders, and Kij represents the observed number of bidders. Nij ≥ Kij as some
potential bidders are not observed and Mij ≥ Kij as bidders are free to bid as many times as
they want. Denote the bids for an auction as bm for m = 1, . . . , Mij and the timing of bids as
tm , for m = 1, . . . , Mij . Also of importance is the timing of the first bid by bidder k denoted
as t1k , for k = 1, . . . , Kij . Bij (t) represents the bidding history at time t which includes all the
(bm , tm , k) pairs previously submitted over [0, t) where k identifies the bidder. Note that while the
timing of the highest proxy bid in Bij (t) is observed the bid amount is not. Knowledge of the bid
history allows one to identify the auction price, sij (t), throughout the auction as well as the timing
at which sij (t) changes. In what follows, we avoid mentioning auction-specific heterogeneity (ij)
except when absolutely necessary.
2.1
The Distribution of Bidders’ Valuations
Bidders’ behavior is modeled in terms of the underlying distribution of bidders’ valuations and the
following two behavioral assumptions.
Assumption 1. Bidders do not bid more than they are willing to pay.
Assumption 2. Bidders do not allow an opponent to win at a price they are willing to beat.
Assumptions 1 and 2 are identical to those in Haile and Tamer (2003). These assumptions
combined with eBay’s proxy bidding system with sufficiently small increments guarantees that
with two or more active bidders the transaction price, s(T ), will equal the highest bid from the
second highest bidder and will also equal that bidder’s valuation. We evaluate the robustness of
this statement in a Supplementary Appendix.
Our assumptions are consistent with many different types of bidding behaviors. Bidders are
free to bid as many times as they want. We only require that the bidder with the second highest
valuation bids her true valuation before the end of the auction and that the bidder with the highest
valuation wins the auction.
5
We account for the presence of heterogeneity across auctions. Denote by (xij , ηij ) the vector
of relevant characteristics describing object j being auctioned by seller i. xij represents the object
characteristics observed by the econometrician and ηij is a real valued index summarizing unobserved object characteristics which is treated as a random effect from a distribution H(·). Denote by
vijk bidder k’s valuation of object j being auctioned by artist i. We impose the following structure:
log(vijk ) = xij 0 β + δηij + wijk , where wijk represents the residual, bidder specific (log-) valuation
of the object being auctioned and is distributed N (0, σ 2 ).
2.2
The Bidders’ Arrival Process
We model only the timing of the first bid by any new bidder and make the following assumption
to identify their arrival.
Assumption 3. Any potential bidder that arrives at an auction at time t immediately places a bid
if their valuation, v, is such that v ≥ s(t).
The process we envision is one in which a bidder arrives to a particular auction, incurs a cost
to learn about the object being auctioned, and then places a bid. This cost may include reading
the auction’s description, looking at available pictures, examining the seller’s reputation and past
auctions, and e-mailing the seller with any questions. After the first bid, we assume the bidder can
continue bidding as many times as she wants at no additional cost.
We model the arrival of new bidders to an auction similar to the arrival of job offers in a
structural job search model (Flinn and Heckman, 1982) and also similar to how others have modeled
the arrival process in an auction environment (Wang, 1993). Denote the instantaneous probability
of arrival of new potential bidders at any time t ∈ [0, T ] as λij (t). Only new potential bidders
with valuation above sij (t) are able to bid which leads to two hazard functions. The instantaneous
probability of arrival of new bidders and the instantaneous probability of arrival of potential,
unrealized bids are
αij (t) = λij (t)(1 − F (sij (t))
γij (t) = λij (t)F (sij (t))
(1)
respectively. The above hazard functions are the product of the arrival rate of potential bidders
and the probability that their valuations are above or below the minimum acceptable bid.
6
Accounting for heterogeneity across auctions, we model the arrival hazard for new potential bidders as log(λij (t)) = log(λ(t)) + xij 0 θ + ϑηij . Our specification of λij (t) includes a baseline hazard
λ(t), an index function xij 0 θ accounting for observed heterogeneity, and an unobserved heterogeneity component ηij , which is treated as a random effect from a distribution H(·). This specification
resembles a mixed proportional hazard model. Mixing with our specification of unobserved heterogeneity accommodates overdispersion and excess zeros (Cameron and Trivedi, 1998).
The likelihood of bidder arrival may not be uniform over the duration of the auction [0, T ].
Impatient bidders may choose to concentrate their attention on auctions close to ending, which the
available search options on eBay facilitates. We address this point by allowing the baseline hazard
λ(t) to take on different values over the duration of the auction. More precisely, we divide [0, T ]
into R mutually exclusive intervals, {Ir }R
r=1 with Ir = [ir−1 , ir ], and allow λ(t) to take on different
values across subintervals while remaining constant within subintervals, log(λ(t)) = λr for t ∈ Ir .
2.3
The Partial Likelihood
The model outlined thus far does not completely specify bidder behavior. Bidders are allowed to
bid and rebid any amount (below their valuation and above the standing bid) as often as they want.
Assumptions 1 and 2 only imply that with two or more bidders the highest bid from the second
highest bidder is that bidder’s valuation, which at the end of the auction will equal the transaction
price. The restriction imposed by Assumption 3 just pertains to the initial time of arrival of any
active bidder.
Estimation of the full likelihood function based on the observed bids and the timing of bids
is not possible given the stated assumptions, as there is an incomplete mapping between bidders’
valuations and all bids, and between bidders’ arrival and the timing of all bids. (We are grateful to
Keisuke Hirano, the Co-Editor, for pointing this out.) Rather than imposing additional restrictions
on bidder behavior, we consider a partial likelihood similar to the approach by Ackerberg, Hirano,
and Shahriar (2006) and noted as an alternative in Villas-Boas (2006). A key difference between
our partial likelihood approach and that by Ackerberg, Hirano, and Shahriar (2006) is that the
probability of observing a particular transaction price is included into our partial likelihood, while
Ackerberg, Hirano, and Shahriar (2006) include this probability via a simulated moment condition.
The partial likelihood for a particular auction in our sample is constructed based on the trans7
action price, s(T ) and the timing of a bidders’ first bid, t1k , (their time of arrival). While the
partial likelihood is a function of all observed bids and the timing of these bids, given our model
assumptions, only the transaction price and the bidders’ time of arrival provide relevant statistical
information. Observed bids and the timing of bids not indicative of arrival play a role in the proper
specification of the bidder arrival process and for this reason are part of the bidding history included
in the conditioning sets.
The partial likelihood function for a particular auction is defined as,
P L (Θ) = P (s(T )|B (tM ), K, tM ; Θ) ·
K
Y
P t1k |B t1k ; Θ
(2)
k=1
with Θ representing the vector of parameters associated to the arrival and bidding processes described in prior subsections, and K is the number of observed bidders. The bidding history, B(t),
includes information from [0, t) and tM is the time the last bid is placed. For notational simplicity,
let P L (Θ) = Ps (Θ) · Pt (Θ) and let B∗ = (B(tM ), K, tM ). The first component of the partial
likelihood is the probability of the transaction price being s(T ) and the second part of the partial
likelihood represents the probability of bidders’ arrival. Observe that P L (Θ) is not a standard
likelihood function because it does not represent the joint density for the observed data, which
would also include the complete vector of realized bids and the timing of additional subsequent
bids.
The second component of the partial likelihood function, Pt (Θ), is defined within the framework
of a multistate duration model, (see, for example, Heckman and Walker (1990)). The multistate
duration model is a generalization of a duration model allowing us to consider several durations
or spells. The probability structure of any duration model can be characterized by its hazard
Rt
function (Lancaster, 1990). For a hazard, λ0 (t), the integrated hazard Λ(t|λ0 ) = 0 λ0 (z)dz, survival
function Ḡ(t|λ0 ) = exp(−Λ(t|λ0 )), distribution function G(t|λ0 ) = 1 − Ḡ(t|λ0 ), and density function
g(t|λ0 ) = λ0 (t)Ḡ(t|λ0 ) are obtained. In Section 2.2, we defined the hazard specifications employed
in our model (equation 1).
The problem of finding an analytic expression for Pt (Θ) is more tractable when the hazard
function is constant. Both α(t) and γ(t), defined in equation (1), vary over the duration of the
auction, [0, T ], from changes in λ(t) (a baseline hazard term) and s(t). We divide the duration of
8
the auction into non-overlapping subintervals such that the hazard functions are constant within
each subinterval in the spirit of Meyer (1990). s(t) changes when new bids are submitted at times
M
{tk }M
k=1 ⊆ [0, T ] and can be divided into M + 1 subintervals with nodes {0, {tk }k=1 , T } where s(t) is
+1
constant within subintervals. Let {Tk }M
k=1 with Tk = [tk−1 , tk ] indicate the subintervals where s(t)
is constant. The baseline hazard term, λ(t), is allowed to take on different values across subintervals
{Ir }R
r=1 in [0, T ] while remaining constant within subintervals. Consider the R + M subintervals of
M +1
the form Dkr ∈ {{Tk ∩ Ir }R
r=1 }k=1 where the hazard functions are constant within a subinterval
Dkr .
The probability Pt (Θ) is determined by analyzing the contribution of each subinterval Dkr .
0 = [a, i ] or D 00 = [a, t ] with a ∈ {i
For each Dkr , there are two scenarios: Dkr
r
r−1 , tk−1 }. In the
k
kr
0 , no new bidder has entered the auction and the contribution to the overall probability
case of Dkr
00 , a new bidder has entered the auction and the
of this subinterval is Ḡ(ir − a|λ0 = α(t)). For Dkr
contribution is g(tk − a|λ0 = α(t)). Finally Pt (Θ) is the product of the corresponding probabilities
from each subinterval Dkr .
The first component of the partial likelihood function, Ps (Θ), represents the contribution of
the transaction price. The conditioning set for this term includes the total number of observed
bidders, K, in the auction. Two special and simple cases are auctions with zero bidders (K = 0)
and one bidder (K = 1) where the transaction price is s0 . For K = 0, none of the potential bidders
valued the object above it’s starting bid and P (s0 |K = 0; Θ) = 1 since this is the only possible bid
sequence with no actual bidders. Similarly for K = 1, P (s0 |K = 1; Θ) = 1 because this is the only
possible outcome from the eBay auction environment with one actual bidder.
When the number of active bidders in an auction is greater than one, the transaction price is
equal to the second highest valuation, s(T ) = v̄N −1 (see Section 2.1). In this case, Ps (Θ) is a
function of the potential number of bidders, N (≥ K), which is unknown. Denote by P (N |B∗ ; Θ)
as the probability that an auction has N potential bidders which depends on the arrival process
defined by Θ. We calculate Ps (Θ) as
P (s(T )|B∗ ; Θ) = EN (P (s(T )|N ; Θ) |B∗ ; Θ)
=
∞
X
P (s(T )|N ; Θ) P (N |B∗ ; Θ) .
N =K
9
(3)
To compute this probability in our empirical application we truncate the infinite sum at the value
N ∗ such that P (N > N ∗ |B∗ ; Θ) < ε. (In our analysis we choose ε = 10−6 and then repeated the
estimation with ε = 10−9 with no significant changes observed in the results.)
For a particular N, the probability P (s(T )|N ; Θ) is derived from the density of the second
order statistic obtained from the underlying distribution of bidders’ valuations, F (v), where
P (s(T )|N ; Θ) = FΘ(N −1:N ) (s(T )) = N (N − 1)fΘ (s(T )) FΘ (s(T ))N −2 (1 − FΘ (s(T ))) .
(4)
Furthermore, the probability of N potential bidders, P (N |B∗ ; Θ) in equation (3) is computed from
the specific bidder arrival process. At each point in time, t, for a given s(t), there are two types
of potential bidders: type I bidders with valuations above s(t) and type II bidders with valuations
below s(t). We observe the arrival of type I bidders and the total number of type I bidders at
the end of the auction is the total number of observed bidders K. The total number of potential
bidders at any time t is the sum of type I and type II bidders and the total number of potential
bidders at the end of the auction is denoted by N . The rate of arrival of type I bidders is governed
by the hazard α(t) and the rate of arrival of type II bidders is governed by the hazard γ(t) both
defined in equation (1). The process that generates N is defined from knowledge of the history of
arrival of type I agents and knowledge of the rate of arrival of type II agents.
M +1
More precisely, consider the R + M subintervals Dkr in {{Tk ∩ Ir }R
r=1 }k=1 where the hazard
functions are constant within subintervals. Within an interval Dkr the arrival of potential type
II bidders, say nkr , follows a simple Poisson process with associated hazard γ(t). When computing P (N |B∗ ; Θ) for a particular (N, K) we need to take into account that this pair imposes a
restriction on the number of type II potential bidders, which must equal N − K. Thus for each
R + M vector of nkr ’s, (n1 , . . . , nR+M ), satisfying the restriction n1 + · · · + nR+M = N − K
we can compute the probability P ((n1 , . . . , nR+M )|N, B∗ ; Θ) as the product of Poisson probabilities P (nkr |γ(t)) = tγ(t)nkr exp−tγ(t) /(nkr !) within each subinterval Dkr .
The sum of all
the P ((n1 , . . . , nR+M )|N, B∗ ; Θ)’s over the set of all possible vectors of (n1 , . . . , nR+M ) defines
P (N |B∗ ; Θ).
Finally, it is also conceptually straightforward within our framework to account for the presence
of an unobserved heterogeneity component, η, with distribution H(η). Unobserved heterogeneity
10
is included in the arrival process and in the bidding process. For each auction, we assume that the
same ηij is associated with both processes. In this sense, we adopt the view that the unobserved
heterogeneity enters the empirical specification like any other variable (size, reputation) and that
the differential impact on the arrival and bidding process is achieved through the multiplicative
parameters.
We incorporate unobserved heterogeneity using the popular semi-parametric approach described
in Heckman and Singer (1984). Subject to the standard assumptions of a random effects model, this
approach hypothesizes a discrete distribution with finite support as a good approximation to the
true distribution of the unobserved heterogeneity component. In this framework, the contribution
to the partial likelihood of an observation can be computed as
P L(Θ) =
H
X
qh P L (Θ, ηh ) ,
h=1
with P L (Θ, ηh ) defined in equation (2). H represents the number of points of support of the
P
discrete distribution. qh ≥ 0 is the probability associated to a realization ηh , with H
h=1 qh = 1.
PH
Identification requires a mean restriction of h=1 qh ηh = 0.
The partial likelihood for the overall sample can be computed as the product the auction specific
partial likelihoods derived throughout this section. The parameter vector, Θ, contains parameters
describing bidders’ valuations and bidders’ arrival process where Θ = (β, δ, λ(·), θ, ϑ, σ, H(·)). After defining the partial likelihood function for the overall sample, standard maximum likelihood
methods can be applied for estimation (Cox 1975, Ackerberg, Hirano, and Shahriar 2006).
3
3.1
Empirical Application
Data Description
From July 13, 2001 to November 14, 2001 we collected all auction data (approximately ten thousand
auctions) from a group of artists, self-denominated EBSQ, who sell their own artwork through eBay.
EBSQ is an abbreviation for “e-basquiat” after the artist Jean-Michel Basquiat. The artists on this
group organize online activities, like art contests. (For more information visit www.ebsqart.com.)
In most cases, the item for sale was an original painting, but other forms of art like collages, ceramic
11
tiles, or sculptures, were also offered. Between August 1, 2004 and December 25, 2004 we conducted
a second round of data collection from a subgroup in the original group of artists consisting of the
artists who listed auctions regularly during the original data collection period. In this study, our
overall sample consists only of original paintings for this subgroup of artists in the years 2001 and
2004. Also, we do not include in our analysis approximately 2% of the auctions in which the ‘buy it
now’ option was exercised. (Before any bids have been submitted, a potential buyer can terminate
the auction and purchase the item being auctioned by paying a ‘buy it now’ price set by the seller.
After the first bid has been submitted, this option disappears.)
In our empirical application we focus our attention on the artists’ choice of auction length
and the effect of this choice on market outcomes. Sellers can choose the auction length from four
possible alternatives: three, five, seven, or ten days. Ten-day auctions require an extra fifty cents
fee. We further restrict our sample to include only seven and ten-day auctions and only artists that
list regularly in both seven and ten-day auctions because we intend to control for artists’ specific
fixed effects. The selected subsample includes auctions by 10 artists from the original group of
37. Even after our restrictions, the selected subsample is still significantly larger than sample sizes
observed in most studies using eBay data. We also employ the overall dataset when empirically
relevant: in particular to conduct tests of the independent private value assumption or to provide
additional support to certain findings from the structural estimation.
For each auction we collected information on item characteristics, auction characteristics, artists’
reputation history, and a complete bidding history, except for the highest bid which is not reported.
Item characteristics include information on the height, width, style (abstract, pop, whimsical, etc.),
medium (acrylic, oil, etc.), and ground (stretch canvas, paper, wood, etc.) of the painting. Auction
characteristics include the length of the auction, the opening bid, the shipping and handling fees,
and the eBay category in which the object was being listed. Each transaction in eBay can be rated
by the buyer and the seller as positive (1), neutral (0) or negative (-1). We collect reputation history
from the artists’ transactions and define two measures of artists’ reputation history: a reputation
index used by eBay representing the percentage of positive feedback, and another measure of artists’
reputation equal to the total number of unique buyers from past transactions. This second measure
of an artist’s reputation is used to proxy the effect of having a large customer base as well as a
potential ‘word of mouth’ effect.
12
Since we model auctions as independent private value second-price ascending-bid auctions, one
might question whether this is the right framework for modeling art auctions. In this regard, the
value of a painting up for auction from the bidder’s perspective is likely to be a combination of
several factors: (1) painting and artist specific characteristics; (2) bidder’s private information about
the painting’s intrinsic value and (3) bidder specific tastes. In our framework the IPV paradigm is
justified if (1) is known a priori by all bidders, (2) plays an immaterial role and (3) is not correlated
across buyers. That is, after controlling for (1) buyers do not value a painting more just because
other potential buyers do. We believe that these conditions hold in our empirical application.
Because the artwork in our sample is auctioned directly from the artist, all bidders have equal
access to a wealth of information about the art auctions. Besides the current auction’s characteristics, bidders also observe the artist’s feedback history as well as information about recently
completed auctions for which links remain active for several weeks after the auction has ended,
including auction characteristics and complete bidding histories. Thus, in general we can discount
a scenario in which one bidder may have private information about the item being auctioned that
could be relevant to other bidders.
We also think it is reasonable to assume that a bidder’s valuation is not affected by other
bidders’ willingness to pay because our data concerns auctions from lesser known artists and this
type of artwork is most likely purchased for personal enjoyment rather than as an investment. In
our view, bidders’ valuations are more likely to be correlated in auctions for artworks by well-known
artists in which case resale can be an attractive option. In this case, bids from other bidders may
convey useful information about the potential future resale value of the artwork. In contrast, most
of the buyers in our sample are unlikely to be very concerned about the resale value of the artwork
and if they choose to resell the artwork on their own they are likely to incur substantial losses
because most buyers in this market would rather buy directly from the artist. It is also highly
unlikely that the artwork in our sample will appreciate in value sufficiently to justify viewing its
purchase as a long term investment. In the highly unlikely case that the artwork does appreciate
substantially over time, the eBay bidding information is unlikely to be relevant. In Section A of
the Appendix we present tests which indicate that the IPV approach is appropriate and further
discuss the relevance of the dataset to our application in the supplementary appendix.
Table 2 presents descriptive statistics for relevant variables from the original sample and the
13
selected subsample. The average number of seven and ten-day auctions among the 10 artists is 59
and 64, respectively. The artists in our subsample experience a higher average probability of sale,
receive bids from a larger number of bidders, and have a more extensive eBay history, as indicated
by the ‘unique feedback’ variable. Auctions with zero bidders represent about 35% of auctions in
the selected subsample, auctions with two, three, or four bidders are not uncommon. Six percent
of the auctions in the seven-day auctions category and 18% in the ten-day auctions had more than
four active bidders. Multiple bids from single bidders are not uncommon. The average size of a
painting is close to two square-feet but there is a large variation in size. All artists’ in our sample
enjoy an excellent eBay feedback rating, with an average rating of about 99.8. The average sale
price is about $52 for seven-day auctions and about $57 for ten-day auctions, including shipping
costs. The observed average number of bidders is 2.3 for seven-day auctions and 3.4 for ten-day
auctions. Other relevant variables not included in this table are style (abstract, contemporary, etc.),
medium (acrylic, pen and ink, etc.), and ground (stretch canvas, canvas panel, etc.). Descriptive
statistics for these additional variables are available from the authors.
3.2
Empirical Framework
We apply the methodology described in Section 2 to the analysis of auction outcomes and to the
artists’ choice of auction length. This application underscores the relevance of a methodology
which considers the process of bidder arrival as well as the distribution of bidders’ valuations.
It also underscores the flexibility of our methodology which can easily be embedded into a more
complex empirical framework.
We model the artists’ choice of a seven-day or ten-day auction and determine how auction
characteristics influence the process of bidder arrival and the distribution of bidders’ valuations.
The artist’s choice of auction length is a potentially complex decision that may depend on the
artist’s expectations about the probability of a sale, the sale price, and the problem of inventory
management, among others. We model this problem using a simple discrete choice framework
where {Iij = 1} represents the ten-day auction choice on the part of the artist and is characterized
as a standard logit model with associated probability pij = L(Ψij ), where L is the logistic CDF,
and with Ψij representing an index function which accounts for artists’ specific fixed effects and
unobserved heterogeneity. (In our empirical specification Ψij takes the form xij 0 ρ + %ηij .) We
14
incorporate the choice of auction length into the model described in Section 2 and the contribution
to the partial likelihood function of a certain auction j from artist i is described as
7
Iij pij P L10
ij + (1 − Iij ) (1 − pij ) P Lij .
7
P L10
ij and P Lij are the partial likelihood components defined by equation (2) and are specific to
ten-day and seven-day auctions.
Our econometric framework postulates an empirical specification governed by four equations:
the artist’s choice of auction length, the process of bidders’ arrival (modeled as two separate Poisson
processes, one for ten-day auctions and one for seven-day auctions), and the bidders’ valuation
of the painting being auctioned, (modeled as a single common equation for seven and ten-day
auctions). We experimented with other model specification structures, like a structure with a single
bidder arrival equation, but this specification was selected based on a likelihood-ratio-test criteria.
Differences across auctions are captured by a vector of covariates as described in Section 3.1. We
also include a dummy for year 2001 that captures any potential overall change in the market across
years, biweekly dummies intended to capture seasonal effects on demand that are common across
years, and a dummy variable intended to capture the potential impact of the terrorists attacks
of 9/11 on auctions that took place in the weeks after this date. The empirical specification also
includes artists’ specific fixed effects to account for unobserved differences across artists, in addition
to those captured by the vector of covariates.
An unobserved heterogeneity component, as described in Section 2, is incorporated into our
empirical specification. We consider a distribution of unobserved heterogeneity with two points of
support and with mean zero and variance one restrictions. Baker and Melino (2000) show that this
specification works well. (The estimated empirical distribution takes values 0.905 and -1.357 with
associated probabilities 0.628 and 0.372.) Estimation results are very similar for models with and
without unobserved heterogeneity, and the coefficients associated to the unobserved heterogeneity
are insignificant in all equations. However, a likelihood ratio test cannot reject the possible presence
of unobserved heterogeneity.
In addition, the process of new bidders’ arrival includes a baseline hazard, λ(·), consisting of
a constant and four additional baseline parameters. The first baseline parameter accounts for
15
differences in the arrival of bidders between the second day and the day before the end of the
auction; the second, third, and fourth baselines account for differences in the rate of arrival on the
last day, the last hour, and the last ten minutes of the auction, respectively. The arrival process
also includes dummies for the day of the week in which the auction ends, allowing for the possible
changes in the flow of potential buyers at different days of the week. Some variables are included
in the bidder-arrival equation but not the bidder-valuation equation. We hypothesize that bidders’
valuations are independent of day-of-the-week effects and other calendar dummies, which are more
likely to affect the process of bidders’ arrival and have no direct effect on bidders’ valuation. These
variables are insignificant when included in the bidders’ valuation equation.
3.3
Estimation Results
Table 3 summarizes the most relevant estimation results. The results from the bidder-valuation
indicate that the most significant determinant of bidder-valuation is painting size, after controlling
for other factors including artists’ fixed effects. An increase in one square foot in size increases
a bidder’s valuation by 30% on average. Interestingly, both measures of seller’s feedback are
significant. An increase in 100 unique previous buyers increases bidders’ valuation by 21%. Possible
interpretations are that this represents a reputation effect or that this measures the increase in
sellers’ intrinsic value as a result of learning-by-doing. Two possible learning-by-doing channels
are improvements in technique or gaining a better understanding of what characteristics of the
artwork bidders’ value the most. In particular, we observe an increase in the average painting size
overtime which is consistent with the artists learning about bidders’ tastes. The average impact of
a decrease in eBay feedback rating of 0.1, for example from 99.8 to 99.7, will result in a relatively
large decrease in average bidders’ valuation of 10.1%.
Our results contrast with reported findings of the cross-sectional literature in this area. Bajari
and Hortaçsu (2003) find that a negative reputation does not significantly impact the final auction
price. Melnik and Alm (2002) estimate that the impact of negative feedback is significant but very
small in magnitude, and the same holds for the impact of the overall rating. Houser and Wooders
(2005) estimate that the average cost to sellers stemming from neutral or negative reputation
scores is less than one percent (0.93%) of the final sales price. Lucking-Reiley, Bryan, Prasad, and
Reeves (2007) estimate that a one percent increase in positive/negative feedback increase/reduce
16
sale price by only 0.03% and 0.11% respectively. Bajari and Hortaçsu (2004) review this evidence
and conclude: “We believe that these results are likely to significantly understate the returns from
having a good reputation...Since getting positive feedback requires effort on the part of sellers, it
appears that sellers are making efforts to avoid negative feedback...” The estimated effect in our
case is much larger than what has been reported in existing cross-sectional studies.
With regards to the bidder-arrival process, we observe again that size has a significant positive
effect. Interestingly, reputation measures have an insignificant impact on bidders’ arrival rates, with
the exception of the ‘number of feedbacks’ variable that is significant for the seven-day bidder-arrival
process. Thus, reputation does not seem to play a significant role in the arrival of bidders for the
group of sellers considered in our analysis. The baseline hazard coefficients associated to the bidderarrival equations are similar across seven and ten-day auctions. The baseline hazard coefficients,
‘Baseline 3’ and ‘Baseline 4’, are larger and more significant indicating that bidders are more likely
to arrive later in the auction. The first and second baselines are somewhat larger for the seven-day
auction arrival process, indicating a lower bidder arrival rate between the first and last day of the
auction in ten-day auctions.
Looking at the equation that models the choice of auction length, both the painting size as
well as the seller reputation measures are significant. We view this as an interesting novel result
which indicates that the measures of seller reputation not only have a significant effect on the
valuations of bidders but also on the actions of the sellers. However, while bidder arrival and
valuation equations are identified from the choices of a large number of independent bidders, the
choice of ten-day auctions is identified from the actions of the sub-group of ten different sellers.
Thus, it is warranted to analyze to what extent the findings about the choice of auction length
also holds in the larger population of artists in the larger dataset. With this purpose in mind we
estimate a conditional logit model in the larger sample with the same specification and controlling
for artist’s specific fixed effects. The results from this exercise are comparable to those from our
structural estimation. The coefficients associated with the first and second measure of reputation
are 3.92 and 0.59, respectively and are significant with associated t-values 2.7 and 4.3 respectively.
These values represent relatively large effects taking into account that the proportion of ten-day
auctions is 27% in the large sample.
17
3.4
Counter-factual Analysis
In addition to examining what factors influence the choice of auction length, we conduct a counterfactual analysis simulating what would happen if seven-day auctions were instead listed as ten-day
auctions. (We thank an anonymous referee for suggesting this application.) This analysis highlights
the value of our structural model and estimates the impact of auction length on market outcomes
such as sellers’ revenue, buyers’ surplus and the revenue of the market intermediary.
The bidders’ arrival component of our structural model allows us to determine the impact of
changes in auction length on the number of potential bidders in an auction. Similarly, the bidders’
valuation component of our structural model is applied, along with the implied changes in the
number of potential bidders, to ascertain the impact of a change in auction length in the final
transaction price. The final transaction price has an impact on sellers’ revenue, the revenue of the
market intermediary and the buyers’ surplus. Changes in the value of these factors as a result of
changes in the auction length can be analyzed within the proposed structural framework.
We conduct our experiment assuming the estimated structural model represents an accurate
description of the data generating process. For each seven-day auction in our selected subsample we generate 100 realizations of potential outcomes resulting from posting an item with the
same characteristics in an auction of a specific length and consider the average outcome from the
simulation.
Simulation results for the average outcome for seven and ten day auctions are presented in
Table 4. The first two columns present results for seven-day auctions, the next two columns present
results under an alternative ten-day scenario, and in the final column we report the difference in
overall outcomes. We observe that the number of potential bidders is similar across auction formats
but about 3% higher for ten-day auctions. This is consistent with our prior results concerning the
bidder’s arrival process in seven and ten day auctions which suggests a higher intensity of bidder
arrival in seven day auctions, especially towards the end of the auction. Average seller revenues are
$52.7 and $57.3 for seven and ten day auctions. The simulated revenues for seven-day auctions are
in line with what we observe in the data but the simulations predict a higher sales rate for seven
day auctions than what we observe in the data (76% predicted vs. 65% observed from Table 2). (In
our analysis we employed several parametric assumptions for simplicity, but our general approach is
18
not reliant on theses assumptions. Thus, the model fit could potentially improve by experimenting
with different semi-parametric techniques and model specifications. This is potentially an avenue
for further research, but beyond the current scope of the paper.) Comparing simulation results for
the second highest and highest bids we compute the average buyer surplus. Buyer surplus is $50
and $52 for seven and ten day auctions, respectively. Finally, examining the list of fees applied
by eBay we approximately compute eBay’s revenues for seven and ten day auctions. Our results
indicate that eBay revenues are $128 (6%) higher for ten-day auctions.
4
Conclusions
This paper presents a novel structural approach for the estimation of independent private value,
ascending-bid, second-price auctions, the most popular type of auction mechanism used on Internet
markets. Unlike previous studies, our estimation approach makes use of information on the time of
arrival of bids, which is readily available on eBay, along with information on bids from all auction
participants. Both the arrival of bidders and the distribution of bidders’ valuations are modeled
explicitly. In addition, we also take into account the specific features of eBay auctions: the number
of potential bidders is a stochastic function of the auction characteristics, and not all potential
bids and bidders are observed. Our approach avoids potential problems of selection bias present in
existing studies.
Estimation is performed by maximizing a partial likelihood function. While our estimated
model includes a number of parametric functional form assumptions, it is possible to use semiparametric model specifications instead. We leave this for future work. We apply our methodology
to the analysis of the artist’s choice of auction length (seven or ten-day auctions) and to the
determination of auction outcomes in general. This application underscores the flexibility of our
methodology which can be easily embedded into a more complex empirical framework.
Our analysis indicates that painting size is an important determinant of bidder’s valuation
and bidder’s participation in an auction. We also consider the effect of two measures of seller’s
reputation and find that reputation has a significant effect on bidders valuations. The estimated
effect is much larger than what has been reported in existing cross-sectional studies. In contrast,
the impact of a seller’s reputation on bidder’s auction participation is to a large extent insignificant.
19
Interestingly when we consider the seller’s choice of auction length, our analysis indicates that
a seller’s reputation has a significant effect on this choice. We find that a seller with a better
reputation is more likely to choose an auction with a longer duration, thus a seller’s reputation
impacts not only the choices of bidders but the choices of sellers as well. Our counter-factual
analysis indicates that if sellers were to list seven day auctions as ten day auctions, the number of
potential bidders would slightly increase and seller revenues would increase by 9%.
20
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22
t
0
1
2
3
j
C
D
A
B
–
–
–
–
–
Table 1: Three Stylized Examples of eBay Auctions
Auction 1
Auction 2
Auction 3
bk
s(t) – CWB
j – bk
s(t) – CWB
j – bk
s(t)
$75
$0.01 – C
B – $30
$0.01 – B
D – $100
$0.01
$100
$76 – D
C – $25
$26 – B
A – $25
$26
?
$76 – D
D – $80
$31 – D
C – $75
$76
?
$76 – D
C – $75
$76 – D
B– ?
$76
–
–
–
–
–
CWB
D
D
D
D
NOTE: Each auction has 4 potential bidders j ∈ {A, B, C, D}, with respective valuations $25, $50,
$75 and $100. Bidders submit bids bk at various points in time t. The current price, s(t), depends
on the 2nd highest bid and the bid increment. Each auction starts at $0.01. The current winning
bidder (CWB) is shown at each point in the auction. A bid of ? indicates that s(t) is above that
bidders valuation. In Auction 2, Bidder A arrives after Bidder C and is unable to bid.
Selected Sample
Square Feet
eBay Feedback
Unique Feedbacks∗
Sale Price∗∗
Bidders∗∗
Bids∗∗
Auctions per Artist
N. Obs. (% Sold)
Range
% Frequency
Overall Sample
Square Feet
eBay Feedback
Unique Feedbacks∗
Sale Price∗∗
Bidders∗∗
Bids∗∗
N. Obs. (% Sold)
Range
% Frequency
Table 2: Descriptive Statistics for Relevant Variables
Seven-Day Auctions
Ten-Day Auctions
Average
Min.
Max.
Average
Min.
1.77
0.01
13.33
1.60
0.02
99.85
99.24
100
99.79
99.24
3.31
0.00
11.53
4.42
0.21
51.87
2.23
349
56.5
6.40
2.30
1
9
3.38
1
3.9
1
23
6.02
1
58.9
27
121
64.4
8
589 (65%)
644 (79%)
0
35.0
Number of Bidders
1
2
3-4 5-6
22.7 19.4 17.2 5.6
Average
2.15
99.89
2.65
41.01
2.13
3.48
0
48.2
Min.
0.01
98.36
0.00
2.23
1
1
2523 (51%)
7+
0.2
Max.
24.00
100
16.50
595.55
9
38
Number of Bidders
1
2
3-4 5-6
21.4 14.7 11.5 3.1
7+
0.6
0
20.2
Number of Bidders
1
2
3-4
5-6
16.8 16.2 28.9 10.9
Average
1.80
99.79
3.64
59.79
3.26
5.98
0
25.1
Max.
20.25
100
11.12
426.02
13
29
228
Min.
0.02
98.44
0.00
6.40
1
1
953 (74%)
7+
7.1
Max.
20.25
100
11.12
426.02
13
29
Number of Bidders
1
2
3-4
5-6
17.7 14.9 25.8 10.3
7+
6.1
NOTE: Sale price includes shipping costs. (∗ ) Measured in hundreds of feedbacks by unique buyers.
(∗∗ ) Refers to sold paintings.
23
Table 3: Estimation Results from Model with Unobserved Heterogeneity
Ten-Day
Bidder Arrival
Bidder Arrival
Bidder
Auction Choice Ten-Day Auction Seven-Day Auction
Valuation
Coef.
t-val.
Coef.
t-val.
Coef.
t-val.
Coef. t-val.
Intercept
-10.39
5.75
-11.73
8.94
-11.62
7.09
-0.343
0.62
Baseline 1
-0.648
8.75
-0.426
4.03
Baseline 2
0.5749
5.53
0.7760
6.15
Baseline 3
2.5117
15.81
2.9360
16.69
Baseline 4
5.3731
45.28
4.9595
29.70
eBay Feedback
3.9270
2.66
-0.013
0.01
0.3009
0.20
1.0149 2.25
Unique Feedbacks
0.5942
4.25
-0.077
0.50
-0.149
1.16
0.2121 5.37
Square Feet
0.3426
2.48
0.1920
3.37
0.2104
2.59
0.3263 10.48
Square Feet2
-0.023
2.05
-0.011
3.11
-0.015
2.63
-0.012
5.11
U. Heterogeneity
-0.110
0.00
0.2010
0.00
-0.128
0.00
0.2829 0.00
Item Dummies
Yes
Yes
Yes
Yes
Artists’ FE
Yes
Yes
Yes
Yes
Calendar Dummies
Yes
Yes
Yes
No
Number of Observations: 1233
Log-likelihood: -34473.21
NOTE: Item Dummies indicate item specific dummies for style, medium and ground types.
Table 4: Simulated Market Outcomes For Two Auction Lengths
Seven-Day Auctions
Ten-Day Auctions
Difference
Average
Overall
Average
Overall
Overall
eBay Revenues
3.39
1997
3.61
2125
128
Buyers’ Surplus
50.32
20952
51.74
22835
1883
Sellers’ Revenues
52.73
21957
57.31
25292
3334
Potential Bidders (% sold)
3.47 (76%)
3.59 (77%)
NOTE: Characteristics of each seven-day auction in our selected subsample are used for simulation.
Simulated values are generated using average results from 100 draws from the estimated model.
Seller revenues represent gross values before fees to eBay.
Table 5: Regression of Bid Value on the Number of Bidders
OLS
IV
Bidders’ FE
Coef.
t-val.
Coef.
t-val.
Coef.
t-val.
Number of Bidders
1.2665
4.02
2.6112
1.84
2.4408
6.81
eBay Feedback
15.719
2.15
12.090
1.50
Unique Feedbacks
0.3908
0.81
0.3797
0.37
Square Feet
15.718
22.17
15.036
14.96
12.038
14.98
2
Square Feet
-0.4281
8.88
-0.3879
6.11
-0.2321
4.43
Number of Obs.
4695
4695
4695
R Squared
0.5177
0.5159
0.8155
NOTE: All models include controls for style, medium, ground and artists’ fixed effects. The last
model also controls for bidders’ fixed effects.
24
A
Tests of Independent Private Values
In the main body of the paper we present qualitative arguments supportive of the IPV assumption
while in this appendix we formally test the IPV vs. common value (CV) assumption. We rely
on tests that have been applied by researchers in prior studies. Existing tests to distinguish IPV
models from CV models are based on the winner’s curse (Paarsch 1992; Haile, Hong and Shum
2003; Bajari and Hortaçsu 2003). In CV auctions, bidders lower their bids in expectation of the
winner’s curse and the amount bids are lowered is a function of the number of observed bidders.
The winner’s curse does not apply to the IPV paradigm where there is no signaling component to
additional observed bidders.
We use IPV/CV tests similar to those used by Bajari and Hortaçsu (2003) and present a novel
test taking advantage of the unique panel structure of our dataset. The tests we use regress bidders’
highest bids on the number of bidders in the auction and other auction characteristics. Under the
CV (IPV) paradigm we would expect to observe a negative (positive) coefficient associated to the
number of bidders variable. (The coefficient for the number of bidders is expected to be positive for
IPV due to the truncation from the eBay auction format. See Footnote 16 of Bajari and Hortaçsu
(2003) and Theorem 9 of Athey and Haile (2002).)
The tests we use distinguish a pure common value model from an IPV model. These tests
do not distinguish between IPV and an interdependent private values model. Haile, Hong and
Shum (2003) develop a test to distinguish between IPV and interdependent values with unobserved
heterogeneity and endogenous entry, but this test is only relevant for first price auctions rather than
the second price auctions considered here. Additionally, in practice it may be difficult to distinguish
an interdependent private values model from a model of unobserved heterogeneity. Given the very
limited role that unobserved heterogeneity plays in the results from our structural model, we feel
that the IPV assumption is not inappropriate for our application.
Table 5 presents results from our IPV/CV tests for the overall sample of auctions with two or
more bidders. The first model presents results from a simple OLS regression that includes controls
for relevant painting characteristics (i.e. style, medium and ground) as well as artist’ reputation
and artists’ fixed effects. Observable characteristics in our sample explain a large proportion of
the variation in final auction prices. Furthermore, our structural analysis suggests that unobserved
heterogeneity plays a very limited role. Thus, we feel confident that the OLS specification of the
test provides relevant information. Furthermore, a second model considers an IV approach that
includes a battery of calendar dummies (i.e. day of the week and month) as instruments for the
number of bidders, which should account for changes in demand but should not affect bidders’
valuations directly, after controlling for the number of bidders’ in an auction. These variables
are insignificant when included in our structural analysis in the bidders’ valuation equation. One
potential problem with the previous two tests is that they do not take into account the potential
effect of differences in bidders’ tastes. To control for this confounding effect one needs a panel data
that includes repeated observations of bids for the same bidders, which we have in our data. The
coefficients associated with seller’s reputation and with painting size have the expected sign for all
models with the exception of the last one which does not include controls for seller’s reputation
because there is no sufficient remaining variation to identify these coefficients with any level of
precision. The coefficient associated to the number of bidders is positive in all models and ranges
from $1.2 to $2.6. This coefficient is significant at the usual level for the first and third model,
while it is significant at the 90% level for the second model. These results are supportive of the
IPV paradigm.
25